For the DFT, all signals and spectra are length . A length sequence
can be denoted by ,
, where may be
real (
) or complex (
). We now wish to regard as a
vector5.1 in an dimensional vector space. That is,
each sample is regarded as a coordinate in that space.
A vector is mathematically a single point in
-space represented by a list of coordinates
called an -tuple. (The
notation means the same thing as .) It can be interpreted
geometrically as an arrow in -space from the origin
to the point
.

We define the following as equivalent:

where
is the th sample of the signal (vector) .
From now on, unless specifically mentioned otherwise, all signals are
length .

Under the geometric interpretation of a length signal, each sample is a
coordinate in the dimensional space. Signals which are only two
samples long are not terribly interesting to hear,5.2 but they are easy to
plot geometrically.